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01.11.2014 | Research Article

Runs, panics and bubbles: Diamond–Dybvig and Morris–Shin reconsidered

verfasst von: Eric Smith, Martin Shubik

Erschienen in: Annals of Finance | Ausgabe 4/2014

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Abstract

The basic two-noncooperative-equilibrium-point model of Diamond and Dybvig is considered along with the work of Morris and Shin utilizing the possibility of outside noise to select a unique equilibrium point. Both of these approaches are essentially nondynamic. We add an explicit replicator dynamic from evolutionary game theory to provide for a sensitivity analysis that encompasses both models and contains the results of both depending on parameter settings.

Vorheriger Artikel Legal enforcement, default and heterogeneity of project-financing contracts

Nächster Artikel Financial soundness indicators and financial crisis episodes

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The replicator dynamic is equivalent to Bayesian learning from repeated samples (Shalizi 2009), so this formulation captures a wide range of cognitive as well as population-level processes for convergence to equilibria.

The simplification of the utility consists of omitting explicit time-discounting structure which increases the appeal of the narrative particular to bank runs but can be replaced by general conditions on the saturation of a single utility of payouts. The continuous variables removed include amounts of the endowment invested, and amounts withdrawn, which in the original model (Diamond and Dybvig 1983) took binary boundary solutions anyway.

Diamond and Dybvig (1983) characterize the surprise signal that one must withdraw early as “private information” to justify the lack of Arrow–Debreu securities within a General Equilibrium framework. The model interpretation extends, however, to many other reasons for incomplete contracts.

The more general case, maximizing ex ante expected utility for agents who are required to deposit, maximizes the function

$$\begin{aligned} \left\langle \theta \right\rangle u \! \left( r \right) + \left( 1 - \theta \right) u \! \left( \frac{ \left( 1 - \left\langle \theta \right\rangle r \right) R }{ \left( 1 - \left\langle \theta \right\rangle \right) } \right) \end{aligned}$$

over $r$. The result is the condition reported in Diamond and Dybvig (1983), Diamond (2007), that

$$\begin{aligned} u^{\prime } \! \left( r \right) = R u^{\prime } \! \left( \frac{ \left( 1 - \left\langle \theta \right\rangle r \right) R }{ \left( 1 - \left\langle \theta \right\rangle \right) } \right) \!. \end{aligned}$$

In Morris and Shin (1998) imperfect observation is discussed in a wider context of formalizing information and beliefs. Here our concern is with the mechanistic consequences of dispersing population responses, which may be given comparable analyses in many contexts of inference, reinforcement learning, or population updating.

Here for brevity we pass over how the negotiation is performed. Because a unique equilibrium value will be computable for the cases we consider, at the equilibrium it will be sufficient for all agents to declare it independently. In a larger discussion of convergence toward equilibrium, many mechanisms could be specified, along lines similar to those we describe below for the selection of threshold strategies, and the convergence of these mechanisms to deposit- or no-deposit-equilibria could be pursued in a more elaborate dynamical model.

Because the replicator dynamic with utility as fitness is equivalent to Bayesian updating (Shalizi 2009), the dynamics described here apply equally well to reinforcement learning by Bayesian updating or to literal population-dynamic processes.

We use the value $r = 1.2$, which is larger than the equilibrium payout value $r \approx 1.1$, because its interesting transitions occur in the center of the $\theta $ range where they are easier to view in graphs. We also use a distribution ${\rho }^{*} \! \left( {\theta }^{*} \right) $ in the first two panels of Fig. 3, to illustrate population structure and the qualitative behavior of relative utility, which is 10x wider than the distribution used to identify the replicator-dynamic velocity and to closely approximate the Nash fixed points in the third panel and later figures.

This usage follows Sewall Wright (1984a, b, c, d).

This restriction on $u$ says that the utility increases more slowly than the logarithm on $1 < r < R$. Combining this requirement with a lower bound $u^{\prime } \! \left( 0 \right) > -\infty $ motivates the form of utility shown in Fig. 2. In the original model of Diamond and Dybvig (1983) the same effect was achieved by discounting period-two payouts in the utility relative to period-one payouts, but apart from some narrative appeal to the case of bank runs, the method used is not critical to the result. The reason such a strong requirement exists on utility, however, is that early withdrawal reduces period-two payout proportionally, to $\left( 1 - f_{\theta } r \right) R$, amplifying the second marginal utility in Eq. (14) by $R$.

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Titel: Runs, panics and bubbles: Diamond–Dybvig and Morris–Shin reconsidered
verfasst von: Eric Smith
Martin Shubik
Publikationsdatum: 01.11.2014
Verlag: Springer Berlin Heidelberg
Erschienen in: Annals of Finance / Ausgabe 4/2014
Print ISSN: 1614-2446
Elektronische ISSN: 1614-2454
DOI: https://doi.org/10.1007/s10436-013-0231-8

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